Since \(ev_{x_{out}}: \mathfrak {M}^{ev}(\mathcal {X},\beta ) \rightarrow X\)  is log smooth, hence is log flat, by definition of transversality and log smooth morphisms being preserved under fine and saturated pullback, the projection \(W \times ^{fs}_X \mathfrak {M}^{ev}(\mathcal {X},\beta ) \rightarrow W\)  is integral and log flat. By [1]} Proposition \(2.3(2)\) , this projection is flat. The equality of log and standard fiber dimension follows from Proposition \(A.7(2)\)  of [1]}.
